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STAN GIBILISCO

McGRAW-HILL

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  To Samuel, Tim, and Tony

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CONTENTS

Preface xiii Acknowledgments xv

CHAPTER 1 Numbering Systems

  1 Sets

  1 Denumerable Number Sets

  6 Bases 10, 2, 8, and 16

  10 Nondenumerable Number Sets

  15 Special Properties of Complex Numbers

  20 Quick Practice

  24 Quiz

  27 CHAPTER 2 Principles of Calculation

  29 Basic Principles

  29 Miscellaneous Principles

  33 Advanced Principles

  37 Approximation and Precedence

  42 Quick Practice

  46 Quiz

  47 CHAPTER 3 Scientific Notation

  51 Powers of 10

  51 Calculations in Scientific Notation

  57 Significant Figures

  61 Quick Practice

  65 Quiz

  67

  CHAPTER 4 Coordinates in Two Dimensions

  71 Cartesian Coordinates

  71 Simple Cartesian Graphs

  74 Polar Coordinates

  80 Navigator’s Coordinates

  87 Coordinate Conversions

  89 Other Coordinate Systems

  92 Quick Practice

  99 Quiz 101

  CHAPTER 5 Coordinates in Three Dimensions 105 Cartesian 3-Space 105 Other 3D Coordinate Systems 108 Hyperspace 113 Quick Practice 119 Quiz 122

  CHAPTER 6 Equations in One Variable 125 Operational Rules 125 Linear Equations 127 Quadratic Equations 130 Higher-Order Equations 134 Quick Practice 137 Quiz 139

  CHAPTER 7 Multivariable Equations 143 2 ×2 Linear Equations 143 3 ×3 Linear Equations 148 2 ×2 General Equations 152 Graphic Solution of Pairs of Equations 154 Quick Practice 158 Quiz

  160

  CHAPTER 8 Perimeter and Area in Two Dimensions 163 Triangles 163 Quadrilaterals 166

  Regular Polygons 171 Circles and Ellipses 172 Other Formulas 175 Quick Practice 180 Quiz

  182

  CHAPTER 9 Surface Area and Volume in Three Dimensions 185 Straight-Edged Objects 185 Cones and Cylinders 191 Other Solids 198 Quick Practice 202 Quiz 204

  CHAPTER 10 Boolean Algebra 207 Operations, Relations, and Symbols 207 Truth Tables 212 Some Boolean Laws 216 Quick Practice 220 Quiz 223

  CHAPTER 11 Trigonometric Functions 227 The Unit Circle 227 Primary Circular Functions 229 Secondary Circular Functions 232 The Right Triangle Model 234 Trigonometric Identities 237 Quick Practice 245 Quiz 248

  CHAPTER 12 Vectors in Two and Three Dimensions 251 Vectors in the Cartesian Plane 251 Vectors in the Polar Plane 256 Vectors in Cartesian 3-Space 259 Standard Form of a Vector 264 Basic Properties 267 Other Properties 275 Quick Practice 278 Quiz 280

  CHAPTER 13 Logarithmic and Exponential Functions 283 Logarithmic Functions 284 How Logarithmic Functions Behave 287 Exponential Functions 290 How Exponential Functions Behave 293 Quick Practice 298 Quiz 300

  CHAPTER 14 Differentiation in One Variable 305 Definition of the Derivative 305 Properties of Derivatives 311 Properties of Curves 315 Derivatives of Wave Functions 323 Quick Practice 329 Quiz 331

  CHAPTER 15 Integration in One Variable 337 What Is Integration? 337 Basic Properties of Integration 341 A Few More Formulas 343 Integrals of Wave Functions 348 Examples of Definite Integration 354 Quick Practice 358 Quiz 361

  Final Exam 365 Answers to Quiz and Exam Questions 395 Suggested Additional Reading 399 Index 401

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  This book is written for people who want to refresh or improve their mathemat- ical skills, especially in fields applicable to science and engineering. The course can be used for self-teaching without the aid of an instructor, but it can also be useful as a supplement in a classroom, tutored, or home-schooling environment. If you are changing careers, and your new work will involve more mathematics than you’ve been used to doing, this book should help you prepare.

  If you want to get the most out of this book, you should have completed high-school algebra, high-school geometry and trigonometry, and a first-year course in calculus. You should be familiar with the concepts of rational, real, and complex numbers, linear equations, quadratic equations, the trigonometric func- tions, coordinate systems, and the differentiation and integration of functions in a single variable.

  This book contains plenty of examples and practice problems. Each chapter ends with a multiple-choice quiz. There is a multiple-choice final exam at the end of the course. The questions in the quizzes and the exam are similar in for- mat to the questions in standardized tests.

  The chapter-ending quizzes are open-book. You may refer to the chapter texts when taking them. When you think you’re ready, take the quiz, write down your answers, and then give your list of answers to a friend. Have the friend tell you your score, but not which questions you got wrong. The answers are listed in the back of the book. Stick with a chapter until you get most, and preferably all, of the quiz answers correct.

  The final exam contains questions drawn uniformly from all the chapters. It is a closed-book test. Don’t look back at the text when taking it. A satisfactory score is at least three-quarters of the answers correct (I suggest you shoot for 90 percent). With the final exam, as with the quizzes, have a friend tell you your score without letting you know which questions you missed. That way, you will not subconsciously memorize the answers. You can check to see where your knowledge is strong and where it is weak.

  I recommend that you complete one chapter a week. An hour or two daily ought to be enough time for this. When you’re done with the course, you can use this book as a permanent reference.

  Suggestions for future editions are welcome.

  TAN

  IBILISCO

  S G

  

  I extend thanks to my nephew Tony Boutelle, a student at Macalester College in St. Paul. He spent many hours helping me proofread the manuscript, and he offered insights and suggestions from the point of view of the intended audience.

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TECHNICAL MATH

  

DEMYSTIFIED

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  This chapter covers the basic properties of sets and numbers. Familiarity with these concepts is important in order to gain a solid working knowledge of applied mathematics. For reference, and to help you navigate the notation you’ll find in this book, Table 1-1 lists and defines the symbols commonly used in tech- nical mathematics.

  

  A set is a collection or group of definable elements or members. A set element can be anything—even another set. Some examples of set elements in applied mathematics and engineering are:

• Points on a line
• Instants in time

  Symbol Description ( ) Quantification; read “ the quantity” [ ] Quantification; used outside ( ) { } Quantification; used outside [ ] { } Braces; objects between them are elements of a set ⇒ Logical implication or “if/then” operation; read “implies” ⇔ Logical equivalence; read “if and only if”

  ∀ Universal quantifier; read “For all” or “For every” ∃ Existential quantifier; read “For some” : Logical expression; read “such that”  Logical expression; read “such that” & Logical conjunction; read “and” ∨ Logical disjunction; read “or” ¬ Logical negation; read “not” N The set of natural numbers Z The set of integers Q The set of rational numbers R The set of real numbers ℵ Transfinite (or infinite) cardinal number ∅ The set with no elements; read “the empty set” or “ the null set” ∩ Set intersection; read “intersect”

• Addition; read “plus” − Subtraction, read “minus” × Multiplication; read “ times” or “multiplied by” ∗ Multiplication; read “ times” or “multiplied by” · Multiplication; read “ times” or “ multiplied by” ÷ Quotient; read “ over” or “divided by”

  ∪ Set union; read “ union” ⊂ Proper subset; read “ is a proper subset of” Symbol Description

  ⊆ Subset; read “is a subset of” ∈ Element; read “is an element of” or “is a mem- ber of” ∉ Nonelement; read “ is not an element of” or “ is not a member of ” = Equality; read “equals” or “is equal to”

  ≠ Not-equality; read “does not equal” or “ is not equal to” ≈ Approximate equality; read “is approximately equal to” < Inequality; read “ is less than” ≤ Equality or inequality; read “ is less than or equal to” > Inequality; read “ is greater than”

  ≥ Equality or inequality; read “ is greater than or equal to”

  / Quotient; read “over” or “divided by” ! Product of all natural numbers from 1 up to a certain value; read “ factorial” × Cross (vector) product of vectors; read “cross”

• Dot (scalar) product of vectors; read “dot” Table 1-1. Symbols commonly used in mathematics.
• Coordinates in a plane. Coordinates in space.

• Points, lines, or curves on a graph.
• Digital logic states.
• Data bits, bytes, or characters.
• Subscribers to a network.
• Wind-velocity vectors at points in the eyewall of a hurricane.
• Force vectors at points along the length of a bridge.

  If an element a is contained in a set A, then the fact is written like this:

  a

  ∈ A

SET INTERSECTION

  The intersection of two sets A and B, written A ∩ B, is the set C consisting of the elements in both sets A and B. The following statement is valid for every element x:

  x

  ∈ C if and only if x ∈ A and x ∈ B

SET UNION

  The union of two sets A and B, written A ∪ B, is the set C consisting of the elements in set A or set B (or both). The following statement is valid for every element x:

  x ∈ C if and only if x ∈ A or x ∈ B

COINCIDENT SETS

  Two nonempty sets A and B are coincident if and only if they are identical. That means that for all elements x, the following statements are both true: If x ∈ A, then x ∈ B If x ∈ B, then x ∈ A

  A B A B Fig. 1-1. The intersection of two non- disjoint, noncoincident sets A and B.

DISJOINT SETS

  Two sets A and B are disjoint if and only if both sets contain at least one element, but there is no element that is in both sets. All three of the following conditions must be met:

  A ≠ ∅ B

  ≠ ∅

  A ∩ B = ∅ where ∅ denotes the empty set, also called the null set.

VENN DIAGRAMS

  The intersection and union of nonempty sets can be conveniently illustrated by

  Venn diagrams . Figure 1-1 is a Venn diagram that shows the intersection of two

  sets that are nondisjoint (they overlap) and noncoincident (they are not identi- cal). Set A ∩ B is the cross-hatched area, common to both sets A and B. Figure 1-2 shows the union of the same two sets. Set A ∪ B is the shaded area, repre- senting elements that are in set A or in set B, or both.

  SUBSETS

  A set A is a subset of a set B, written A ⊆ B, if and only if any element x in set

  A is also in set B. The following logical statement holds true for all elements x:

  If x ∈ A, then x ∈ B

  A B A B Fig. 1-2. The union of two non-disjoint, noncoincident sets A and B.

PROPER SUBSETS

  A set A is a proper subset of a set B, written A ⊂ B, if and only if any element x in set A is in set B, but the two sets are not coincident. The following logical statements both hold true for all elements x:

  If x ∈ A, then x ∈ B

  A ≠ B CARDINALITY

  The cardinality of a set is the number of elements in the set. The null set has zero

  cardinality . The set of data bits in a digital image, stars in a galaxy, or atoms in

  a chemical sample has finite cardinality. Some number sets have denumerably

  infinite cardinality . Such a set can be fully defined by a listing scheme. An

  example is the set of all counting numbers {1, 2, 3, . . . }. Not all infinite sets are denumerable. There are some sets with non-denumerably infinite cardinality. This kind of set cannot be fully defined in terms of any listing scheme. An example is the set of all real numbers, which are those values that represent mea- surable physical quantities (and their negatives).

PROBLEM 1-1

  Find the union and the intersection of the following two sets:

  S = {2, 3, 4, 5, 6} T

  = {4, 5, 6, 7, 8}

SOLUTION 1-1

  The union of the two sets is the set S ∪ T consisting of all the elements in one or both of the sets S and T. It is only necessary to list an element once if it happens to be in both sets. Thus:

  S

  ∪ T = {2, 3, 4, 5, 6, 7, 8} The intersection of the two sets is the set S ∩ T consisting of all the ele- ments that are in both of the sets S and T:

  S

  ∩ T = {4, 5, 6}

PROBLEM 1-2

  In Problem 1-1, four sets are defined: S, T, S ∪ T, and S ∩ T. Are there any cases in which one of these sets is a proper subset of one or more of the others? If so, show any or all examples, and express these exam- ples in mathematical symbology.

  SOLUTION 1-2 Set S is a proper subset of S ∪ T. Set T is also a proper subset of S ∪ T.

  We can write these statements formally as follows:

  S ⊂ (S ∪ T)

T

  ⊂ (S ∪ T) It also turns out, in the situation of Problem 1-1, that the set S ∩ T is a proper subset of S, and the set S ∩ T is a proper subset of T. In formal symbology, these statements are:

  (S ∩ T) ⊂ S (S ∩ T) ⊂ T

  The parentheses are included in these symbolized statements in order to prevent confusion as to how they are supposed to be read. A mathematical purist might point out that, in these examples, parenthe- ses are not necessary, because the meanings of the statements are evi- dent from their context alone.

  

  The set of familiar natural numbers, the set of integers (natural numbers and their negatives, including 0), and the set of rational numbers are examples of

  sets with denumerable cardinality. This means that they can each be arranged in the form of an infinite (open-ended) list in which each element can be assigned a counting number that defines its position in the list.

  NATURAL NUMBERS

  The natural numbers, also known as whole numbers, are built up from a starting point of 0. The set of natural numbers is denoted N, and is commonly expressed like this:

  N = {0, 1, 2, 3,..., n,...}

  In some texts, zero is not included, so the set of natural numbers is defined as follows:

  N = {1, 2, 3, 4,..., n,...}

  This second set, starting with 1 rather than 0, is sometimes called the set of counting numbers .

  The natural numbers can be expressed as points along a horizontal half-line or ray, where quantity is directly proportional to displacement (Fig. 1-3). In the illustration, natural numbers correspond to points where hash marks cross the ray. Increasing numerical values correspond to increasing displacement toward the right. Sometimes the ray is oriented vertically, and increasing values correspond to displacement upward.

  INTEGERS

  The set of natural numbers can be duplicated and inverted to form an identical, mirror-image set: −N = {0, −1, −2, −3,..., −n,...}

  Numerical value is proportional to displacement

  1

  2

  3

  4

  5

  

6

  7

  8

  9 Fig. 1-3. The natural numbers can be depicted as discrete points on a half- line or ray. The numerical value is directly proportional to the displacement.

  The union of this set with the set of natural numbers produces the set of inte- gers, commonly denoted Z:

  Z = N ∪ −N = {..., −n,..., −2, −1, 0, 1, 2,..., n,...}

  Integers can be expressed as points along a horizontal line, where positive quantity is directly proportional to displacement toward the right, and negative quantity is directly proportional to displacement toward the left (Fig. 1-4). In the illustration, integers correspond to points where hash marks cross the line. Sometimes a vertical line is used. In most such cases, positive values correspond to upward displacement, and negative values correspond to downward displace- ment. The set of natural numbers is a proper subset of the set of integers. Stated symbolically:

  N ⊂ Z

RATIONAL NUMBERS

  A rational number (the term derives from the word ratio) is a number that can be expressed as, or reduced to, the quotient of two integers, a and b, where b is positive. The standard form for a rational number r is:

  r = a/b

  The set of all possible quotients of this form composes the entire set of rational numbers, denoted Q. Thus, we can write:

  Q = {x | x = a/b, where a ∈ Z, b ∈ Z, and b > 0} “Center” of string of points Negative integers Positive integers

  −8 −6 −4 −2

  2

  4

  6

  8 Fig. 1-4. The integers can be depicted as discrete points on a horizontal line. Displacement to the right corresponds to positive values, and displacement to the left corresponds to negative values. The set of integers is a proper subset of the set of rational numbers. The natural numbers, the integers, and the rational numbers have the following relationship:

  N ⊂ Z ⊂ Q

DECIMAL EXPANSIONS

  Rational numbers can be denoted in decimal form as an integer followed by a period (radix point, also called a decimal point), and then followed by a sequence of digits. The digits to the right of the radix point always exist in either of two forms:

• A finite string of digits beyond which all digits are zero.
• An infinite string of digits that repeat in cycles.

  Here are two examples of the first form, known as terminating decimal

  numbers : 3/4 = 0.750000...

  −9/8 = −1.1250000... Here are two examples of the second form, known as nonterminating, repeat-

  ing decimal numbers :

  1/3 = 0.33333... −123/999 = −0.123123123...

PROBLEM 1-3

  Of what use are negative numbers? How can you have a quantity smaller than zero? Isn’t that like having less than none of something?

SOLUTION 1-3

  Negative numbers are surprisingly common. Most people have experi- enced temperature readings that are “below zero,” especially if the Celsius scale is used. Sometimes, driving in reverse instead of in forward gear is considered to be “negative velocity.” Some people carry a “negative bank balance” for a short time. The government always seems to have a “deficit,” and corporations often operate “in the red.”

  PROBLEM 1-4 Express the number 2457/9999 as a nonterminating, repeating decimal. SOLUTION 1-4

  If you have a calculator that displays plenty of digits (the scientific- mode calculator in Windows XP is excellent), you can find this easily: 2457/9999 = 0.245724572457.... The sequence of digits 2457 keeps repeating “forever.” Note that this number is rational because it is the quotient of two integers, even though it is not a terminating decimal. That is, it can’t be written out fully in deci- mal form using only a finite number of digits to the right of the radix point.

  

  The numbering system used by people (as opposed to computers and calcu- lators) in everyday life is the decimal number system, based on powers of 10. Machines, in contrast, generally perform calculations using numbering systems based on powers of 2.

  DECIMAL NUMBERS The decimal number system is also called modulo 10, base 10, or radix 10.

  Digits are elements of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. The digit immediately to the left of the radix (decimal) point is multiplied by 10 , or 1. The next digit

  1

  to the left is multiplied by 10 , or 10. The power of 10 increases as you move further to the left. The first digit to the right of the radix point is multiplied by a

  −1 −2 factor of 10 , or 1/10. The next digit to the right is multiplied by 10 , or 1/100.

  This continues as you go further to the right. Once the process of multiplying each digit is completed, the resulting values are added. This is what is repre- sented when you write a decimal number. For example:

  3

  2

  1

  2704.53816 = (2 × 10 ) + (7 × 10 ) + (0 × 10 ) + (4 × 10 )

  −1 −2 −3 −4 −5

  ) ) ) ) )

• (5 × 10 + (3 × 10 + (8 × 10 + (1 × 10 + (6 × 10 The parentheses are added for clarity.

BINARY NUMBERS

  The binary number system is a method of expressing numbers using only the digits 0 and 1. It is sometimes called modulo 2, base 2, or radix 2. The digit immediately to the left of the radix point is the “ones” digit. The next digit to the left is a “twos” digit; after that comes the “fours” digit. Moving further to the left, the digits represent 8, 16, 32, 64, and so on, doubling every time. To the right of the radix point, the value of each digit is cut in half again and again, that is, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, and so on.

  Consider an example using the decimal number 94:

  1

  94 ) ) = (4 × 10 + (9 × 10

  In the binary number system the breakdown is:

  

1

  2

  1011110 = (0 × 2 ) + (1 × 2 ) + (1 × 2 )

  3

  4

  5

  6

  ) ) ) )

• (1 × 2 + (1 × 2 + (0 × 2 + (1 × 2 When you work with a computer or calculator, you give it a decimal number that is converted into binary form. The computer or calculator does its operations with zeros and ones, which are represented by different voltages or signals in electronic circuits. When the process is complete, the machine converts the result back into decimal form for display.

OCTAL NUMBERS

  Another numbering scheme, called the octal number system, has eight symbols,

  3

  or 2 cubed (2 ). It is also called modulo 8, base 8, or radix 8. Every digit is an element of the set {0, 1, 2, 3, 4, 5, 6, 7}. Counting thus proceeds from 7 directly to 10, from 77 directly to 100, from 777 directly to 1000, and so on. There are no numerals 8 or 9. In octal notation, decimal 8 is expressed as 10, and decimal 9 is expressed as 11.

HEXADECIMAL NUMBERS

  Yet another scheme, commonly used in computer practice, is the hexadecimal

  number system , so named because it has 16 symbols, or 2 to the fourth power

  4

  (2 ). These digits are the usual 0 through 9 plus six more, represented by A

COMPARISON OF VALUES

  )

  Solving a problem like this is straightforward, but the steps are tricky, tedious, and repetitive. Some calculators will perform conversions like this directly, but if you don’t have access to one, you can proceed in the following manner.

  PROBLEM 1-6 Express the decimal number 1,000,000 in hexadecimal form. SOLUTION 1-6

  ) = (1 × 1) + (1 × 2) + (0 × 4) + (1 × 8)

  7

  )

  6

  )

  5

  4

  through F, the first six letters of the alphabet. The digit set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. In this number system, A is the equivalent of decimal 10, B is the equivalent of decimal 11, C is the equivalent of decimal 12, D is the equivalent of decimal 13, E is the equivalent of decimal 14, and F is the equivalent of decimal 15. This system is also called modulo 16, base 16, or radix 16 .

  )

  3

  ) + (1 × 2

  2

  ) + (0 × 2

  1

  Working from right to left, the digits add up as follows: 10011011 = (1 × 2 ) + (1 × 2

  PROBLEM 1-5 Express the binary number 10011011 in decimal form. SOLUTION 1-5

  In Table 1-2, numerical values are compared in modulo 10 (decimal), 2 (binary), 8 (octal), and 16 (hexadecimal), for the decimal numbers 0 through 64. In gen- eral, as the modulus (or number base) increases, the numeral representing a given value becomes “smaller.”

• (1 × 2
• (0 × 2
• (0 × 2
• (1 × 2
• (1 × 16) + (0 × 32) + (0 × 64) + (1 × 128) = 1 + 2 + 0 + 8 + 16 + 0 + 0 + 128 = 155

  

Decimal Binary Octal Hexadecimal Decimal Binary Octal Hexadecimal

33 100001

  65

  38 24 11000

  70

  17 56 111000

  27

  37 23 10111

  67

  16 55 110111

  26

  36 22 10110

  66

  15 54 110110

  25

  35 21 10101

  14 53 110101

  18 57 111001

  24

  34 20 10100

  64

  13 52 110100

  23

  33 19 10011

  63

  12 51 110011

  22

  32 18 10010

  62

  11 50 110010

  21

  30

  71

  61

  75

  40

  40 32 100000

  1F 64 1000000 100

  37

  3F 31 11111

  77

  1E 63 111111

  36

  3E 30 11110

  76

  1D 62 111110

  35

  3D 29 11101

  1C 61 111101

  39 25 11001

  34

  3C 28 11100

  74

  1B 60 111100

  33

  3B 27 11011

  73

  1A 59 111011

  32

  3A 26 11010

  72

  19 58 111010

  31

  31 17 10001

  10 49 110001

  40

  11

  6

  26 6 110

  46

  5 38 100110

  5

  25 5 101

  45

  4 37 100101

  4

  24 4 100

  44

  3 36 100100

  3

  3

  47

  23

  43

  2 35 100011

  2

  10

  2

  22

  42

  1 34 100010

  1

  1

  1

  21

  6 39 100111

  27 7 111

  20

  54

  30 16 10000

  60

  17 F 48 110000

  2F 15 1111

  57

  16 E 47 101111

  2E 14 1110

  56

  15 D 46 101110

  2D 13 1101

  55

  14 C 45 101101

  2C 12 1100

  13 B 44 101100

  7

  2B 11 1011

  53

  12 A 43 101011

  2A 10 1010

  52

  9 42 101010

  11

  29 9 1001

  51

  8 41 101001

  10

  28 8 1000

  50

  7 40 101000

  20 Table 1-2. Comparison of numerical values for decimal numbers 0 through 64. The values of the digits in a whole (that is, nonfractional) hexadecimal num- ber, proceeding from right to left, are natural-number powers of 16. That means a whole hexadecimal number n has this form:

  16

  

5

  4

  3 n =... + (f × 16 ) + (e × 16 ) + (d × 16 )

  16

  

2

  1

  ) ) )

• (c × 16 + (b × 16 + (a × 16 where a, b, c, d, e, f, . . . are single-digit hexadecimal numbers from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}.

  In order to find the hexadecimal value of decimal 1,000,000, first find the

  4 largest power of 16 that is less than or equal to 1,000,000. This is 16 = 65,536.

  Then, divide 1,000,000 by 65,536. This equals 15 and a remainder. The decimal 15 is represented by the hexadecimal F. We now know that the decimal number 1,000,000 looks like this in hexadecimal form:

  4

  3

  2

  1

  (F × 16 ) + (d × 16 ) + (c × 16 ) + (b × 16 ) + a = Fdcba

  4 In order to find the value of d, note that 15 × 16 = 983,040. This is 16,960

  smaller than 1,000,000. That means we must find the hexadecimal equivalent of decimal 16,960 and add it to hexadecimal F0000. The largest power of 16 that

  3

  is less than or equal to 16,960 is 16 , or 4096. Divide 16,960 by 4096. This equals 4 and a remainder. We now know that d = 4 in the above expression, so decimal 1,000,000 is equivalent to the following in hexadecimal form:

  4

  3

  2

  1

  (F × 16 ) + (4 × 16 ) + (c × 16 ) + (b × 16 ) + a = F4cba

  4

  3 In order to find the value of c, note that (F × 16 ) + (4 × 16 ) = 983,040 +

  16,384 = 999,424. This is 576 smaller than 1,000,000. That means we must find the hexadecimal equivalent of decimal 576 and add it to hexadecimal

  2 F4000. The largest power of 16 that is less than or equal to 576 is 16 , or 256.

  Divide 576 by 256. This equals 2 and a remainder. We now know that c = 2 in the above expression, so decimal 1,000,000 is equivalent to the following in hexadecimal form:

  4

  3

  2

  1

  (F × 16 ) + (4 × 16 ) + (2 × 16 ) + (b × 16 ) + a = F42ba

  4

  3

  2 In order to find the value of b, note that (F × 16 ) + (4 × 16 ) + (2 × 16 ) =

  983,040 + 16,384 + 512 = 999,936. This is 64 smaller than 1,000,000. That means we must find the hexadecimal equivalent of decimal 64 and add it to hexadecimal F4200. The largest power of 16 that is less than or equal to 64 is

  1

  16 , or 16. Divide 64 by 16. This equals 4 without any remainder. We now know that b = 4 in the above expression, so decimal 1,000,000 is equivalent to hexadecimal:

  4

  3

  2

  1 There was no remainder when we found b. Thus, all the digits to the right of

  b (in this case, that means only the digit a) are equal to 0. Decimal 1,000,000 is therefore equivalent to hexadecimal F4240.

  Checking, note that the hexadecimal F4240 breaks down as follows when converted to decimal form, proceeding from right to left:

  1

  

2

  3

  4 F4240 = (0 × 16 ) + (4 × 16 ) + (2 × 16 ) + (4 × 16 ) + (15 × 16 )

  = 64 + 512 + 16,384 + 983,040 = 1,000,000

  

  A number set is nondenumerable if and only if there is no way that its elements can be arranged as a list, where each element is assigned a counting number defining its position in the list. Examples of nondenumerable number sets include the set of irrational numbers, the set of real numbers, the set of imagi-

  nary numbers , and the set of complex numbers. These types of numbers are used to express theoretical values in science and engineering.

IRRATIONAL NUMBERS

  An irrational number cannot be expressed as the ratio of two integers. Examples of irrational numbers include:

• the length of the diagonal of a square that is 1 unit long on each edge (the square root of 2, roughly equal to 1.41421)
• the circumference-to-diameter ratio of a circle in a plane (commonly known as pi and symbolized π, roughly equal to 3.14159)

  Irrational numbers are inexpressible in decimal-expansion form. When an attempt is made to express such a number in this form, the result is a decimal expression that is nonterminating and nonrepeating. No matter how many digits are specified to the right of the radix point, the expression is always an approx- imation, never the exact value.

  The set of irrational numbers can be denoted S. This set is entirely disjoint from the set of rational numbers:

  This means that no rational number is irrational, and no irrational number is rational.

REAL NUMBERS

  The set of real numbers, denoted R, is the union of the sets of rational and irra- tional numbers:

  R = Q ∪ S

  For practical purposes, R can be denoted as the set of points on a continuous geometric line, as shown in Fig. 1-5. (In theoretical mathematics, the assertion that the points on a geometric line correspond one-to-one with the real numbers is known as the Continuum Hypothesis.) The real numbers are related to the rational numbers, the integers, and the natural numbers as follows:

  N ⊂ Z ⊂ Q ⊂ R

  The operations of addition, subtraction, multiplication, division, and expo- nentiation can be defined over the set of real numbers. If # represents any one of these operations and x and y are elements of R, then:

  x # y ∈ R The only exception to this is that for division, y must not be equal to 0.

  Division by 0 is not defined within the set of real numbers.

TRANSFINITE CARDINAL NUMBERS

  The cardinal numbers for infinite sets are denoted using the uppercase aleph ( ℵ), the first letter in the Hebrew alphabet. The cardinality of the sets of natural

  “Center” of continuous line Negative real numbers Positive real numbers −8 −6 −4

  2

  4

  6

  8 −2 Fig. 1-5. The real numbers can be depicted as all the points on a continuous, solid, horizontal line. Displacement to the right corresponds to positive values, numbers, integers, and rational numbers is called ℵ (aleph null, aleph nought, or aleph 0). The cardinality of the sets of irrational and real numbers is called ℵ (aleph one or aleph 1). These two quantities, ℵ and ℵ , are known as trans-

  1

  1 finite cardinal numbers . They are expressions of “infinity.”

  Around the year 1900, the German mathematician Georg Cantor proved that ℵ and ℵ are not the same. This reflects the fact that the elements of the set

  1

  of natural numbers can be paired off one-to-one with the elements of the sets of integers or rational numbers, but not with the elements of the sets of irrational numbers or real numbers. Any attempt to pair off the elements of N with the ele- ments of S, or the elements of N and the elements of R, results in some elements of S or R being “left over” without corresponding elements in N. A simplistic, but interesting, way of saying this is that there are at least two “infinities,” and they are not equal to each other!

IMAGINARY NUMBERS

  The set of real numbers, and the operations defined above for the integers, give rise to some expressions that do not behave as real numbers. The best known example is the quantity j such that j × j = −1. Thus, j is equal to the positive square root of −1. No real number has this property. This quantity j is known as the unit imaginary number or the j operator. Sometimes, in theoretical mathe- matics, j is denoted i.